I know that for measurable functions $f,g:X\rightarrow \mathbb{R}$ that if $f^2$ and $g^2$ are lebesgue integrable then $fg$ is lebesgue integrable (for a finite measure space $X$)
I understand the proof of that and so I’m trying to mimic the idea to prove $fg^2$ is lebesgue integrable provided $f^3$ and $g^5$ are lebesgue integrable.
From my understanding it will be sufficient to show $f^2$ and $g^4$ are lebesgue integrable but I’m not sure how that is done. Thus I’ve tried replicating the proof by using both $(f+g)^5$ and $(f-g)^5$ to show $|fg^2|$ is bounded but there’s more unwanted terms involved than in the proof so I’m not struggling to proceed.
$f^3$ and $g^3$ need to be integrable. Since integrability of $fg^2$ is equivalent to integrability of $|fg^2|$, we can assume that $f$ and $g$ are nonnegative. Then $$ 3fg^2 \le (f+g)^3 $$ and everything is fine.
You can use Hoelder inequality next time: in order that a product of $k$ functions is integrable, it suffices that the $k$-th power of each function is integrable.